Ask HN: Independent Math Study

#1

Post by devin » Sun, Jul 19, 2009, 5:32 AM UTC

I never thought I'd be asking this. Math was never my "strong suit", but over the last year I've really grown to enjoy it as I learn more. I've taken Calc. I in a fairly demanding college environment, and am planning on continuing with Calc. II and Linear Algebra.

My question to HN is: How does one go about doing self-study in Math? It seems, of all the sciences, to be especially difficult to tackle without the built-in support of the classroom. I assume that like most things, it just takes a lot of hard work and study, but I'm curious if anyone out there has a rough plan for tackling a reasonably rich understanding of mathematics on their own. Sites, materials, etc. are appreciated.

Thanks!

Re: Ask HN: Independent Math Study

#2

Post by rms » Sun, Jul 19, 2009, 5:34 AM UTC

I don't have a complete answer for you, but I linked to this book a few days ago. It's pretty good. http://www.math.wisc.edu/~keisler/calc.html

Elementary Calculus: An Infinitesimal Approach for a mathematically rigorous course in infinitesimal calculus. I think it is much more intuitive than typical limit calculus.

Re: Ask HN: Independent Math Study

#3

Post by devin » Sun, Jul 19, 2009, 5:42 AM UTC

rms wrote: post #2

I don't have a complete answer for you, but I linked to this book a few days ago. It's pretty good. http://www.math.wisc.edu/~keisler/calc.html Elementary Calculus: An Infinitesimal Approach for a mathematically rigorous course in infinitesimal calculus. I think it is much more intuitive than typical limit calculus.

Fancy that! I'm at the UW right now. I have been considering a few different routes. Philosophy, Classics, Music, Computer Science, or Math. I'd really like to go for a Math undergrad with a minor in one of the other subjects, but I'm going to need a supplement during the summer to catch up with some of the other math guys at the UW. Lots of competition.

Re: Ask HN: Independent Math Study

#4

Post by jibiki » Sun, Jul 19, 2009, 5:57 AM UTC

rms wrote: post #2

I don't have a complete answer for you, but I linked to this book a few days ago. It's pretty good. http://www.math.wisc.edu/~keisler/calc.html Elementary Calculus: An Infinitesimal Approach for a mathematically rigorous course in infinitesimal calculus. I think it is much more intuitive than typical limit calculus.

There are vast sections of mathematics which cannot be understood without first understanding limits. There are very few areas of mathematics which require understanding infinitesimals.

Re: Ask HN: Independent Math Study

#5

Post by rms » Sun, Jul 19, 2009, 6:18 AM UTC

jibiki wrote: post #4

rms wrote: post #2

I don't have a complete answer for you, but I linked to this book a few days ago. It's pretty good. http://www.math.wisc.edu/~keisler/calc.html Elementary Calculus: An Infinitesimal Approach for a mathematically rigorous course in infinitesimal calculus. I think it is much more intuitive than typical limit calculus.

There are vast sections of mathematics which cannot be understood without first understanding limits. There are very few areas of mathematics which require understanding infinitesimals.

Sure. I just think the infinitesimal is an interesting approach.

Re: Ask HN: Independent Math Study

#6

Post by mlLK » Sun, Jul 19, 2009, 6:26 AM UTC

Checkout this thread, http://news.ycombinator.com/item?id=108723

Re: Ask HN: Independent Math Study

#7

Post by mark-t » Sun, Jul 19, 2009, 6:30 AM UTC

To be honest, calculus isn't that important for mathematicians, but if you want to study mathematics seriously, I'd suggest picking up a rigorous text like Rudin's or Apostol's. It will be difficult. You'll have to read most of it several times. That's perfectly fine; the point is that it will help you learn to think like a mathematician does.

Now, on the other hand, linear algebra is almost universally important and is probably easier for a programmer to grasp. I would also suggest picking up a Number Theory or Combinatorics text; they're practically useless, but they're fun and interesting, they'll give you a better idea of what mathematicians do, and you don't need much education to get into them.

My usual advice for building skills is to work on contest problems. See if you can find some AMC12 problems. If those are too easy, you can work your way up. AIME and Putnam would be good next steps (those can be found here: http://web.archive.org/web/20080205091131/http://www.kalva.d... ).

Re: Ask HN: Independent Math Study

#8

Post by aneesh » Sun, Jul 19, 2009, 6:48 AM UTC

mark-t wrote: post #7

To be honest, calculus isn't that important for mathematicians, but if you want to study mathematics seriously, I'd suggest picking up a rigorous text like Rudin's or Apostol's. It will be difficult. You'll have to read most of it several times. That's perfectly fine; the point is that it will help you learn to think like a mathematician does. Now, on the other hand, linear algebra is almost universally important and…

Saying the Putnam is a next step from AMC12 problems is like saying the NBA is a next step from pickup basketball with friends in middle school! There are people who can do Putnam problems for fun, but those people generally know who they are already.

Re: Ask HN: Independent Math Study

#9

Post by TriinT » Sun, Jul 19, 2009, 6:50 AM UTC

mark-t wrote: post #7

To be honest, calculus isn't that important for mathematicians, but if you want to study mathematics seriously, I'd suggest picking up a rigorous text like Rudin's or Apostol's. It will be difficult. You'll have to read most of it several times. That's perfectly fine; the point is that it will help you learn to think like a mathematician does. Now, on the other hand, linear algebra is almost universally important and…

I would claim that Calculus isn't that important for engineers / scientists / programmers either. Real Analysis is important if one needs to understand thing deeper. In the real world, problems can't be solved analytically... and many of the tools one learns in Calculus are kind of useless. I think Linear Algebra is much, much more important than Calculus. Linear Algebra is the arithmetic of higher mathematics, like Bellman said.

Re: Ask HN: Independent Math Study

#10

Post by sdesol » Sun, Jul 19, 2009, 6:56 AM UTC

It's been ages since my last university math class (I was a I math major), so I can't point you to any reference material, but I can say the following.

If you really want to improve your problem solving skills, I would highly recommend studying real analysis. What you get out of this will go a long way to making you a better problem solver. The reason why I say this is when you have to so something like prove why 1 is greater than 0, you'll learn to look at things differently.

In studying real analysis, you are almost learning how to walk again. Everything that you have taken for granted as being obvious in the past will now have to be proven. And by going through these exercises, you'll learn the importance of truly understanding what you are doing.